| L(s) = 1 | + 2·3-s + 4·7-s + 9-s − 11-s + 4·13-s − 4·19-s + 8·21-s + 6·23-s − 4·27-s − 6·29-s + 8·31-s − 2·33-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 6·47-s + 9·49-s + 6·53-s − 8·57-s − 12·59-s + 2·61-s + 4·63-s + 10·67-s + 12·69-s − 12·71-s + 16·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.917·19-s + 1.74·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s + 1.43·31-s − 0.348·33-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + 1.22·67-s + 1.44·69-s − 1.42·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.791104646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.791104646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.652979243971769604275134320618, −8.751180023146231719351647431684, −8.300774525279624036970758438928, −7.73981333382224107078881372013, −6.62440898627967000941825237421, −5.45587338755955191202866606222, −4.53025089815929260508437447370, −3.55417269649287765468102348375, −2.45503079564260710600733287182, −1.44085657982484826185126926519,
1.44085657982484826185126926519, 2.45503079564260710600733287182, 3.55417269649287765468102348375, 4.53025089815929260508437447370, 5.45587338755955191202866606222, 6.62440898627967000941825237421, 7.73981333382224107078881372013, 8.300774525279624036970758438928, 8.751180023146231719351647431684, 9.652979243971769604275134320618
